Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $r \neq 0$. $t = \dfrac{3r^2 - 21r}{-9r + 18} \times \dfrac{-4r + 8}{r^2 - 5r - 14} $
Solution: First factor the quadratic. $t = \dfrac{3r^2 - 21r}{-9r + 18} \times \dfrac{-4r + 8}{(r - 7)(r + 2)} $ Then factor out any other terms. $t = \dfrac{3r(r - 7)}{-9(r - 2)} \times \dfrac{-4(r - 2)}{(r - 7)(r + 2)} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ 3r(r - 7) \times -4(r - 2) } { -9(r - 2) \times (r - 7)(r + 2) } $ $t = \dfrac{ -12r(r - 7)(r - 2)}{ -9(r - 2)(r - 7)(r + 2)} $ Notice that $(r - 2)$ and $(r - 7)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ -12r\cancel{(r - 7)}(r - 2)}{ -9(r - 2)\cancel{(r - 7)}(r + 2)} $ We are dividing by $r - 7$ , so $r - 7 \neq 0$ Therefore, $r \neq 7$ $t = \dfrac{ -12r\cancel{(r - 7)}\cancel{(r - 2)}}{ -9\cancel{(r - 2)}\cancel{(r - 7)}(r + 2)} $ We are dividing by $r - 2$ , so $r - 2 \neq 0$ Therefore, $r \neq 2$ $t = \dfrac{-12r}{-9(r + 2)} $ $t = \dfrac{4r}{3(r + 2)} ; \space r \neq 7 ; \space r \neq 2 $